math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 96.7%

Time: 6.2s

Alternatives: 8

Speedup: 1.7×

• Unsound rule application detected in e-graph. Results may not be sound.

• 45.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+
  (* (- (* x.re x.re) (* x.im x.im)) x.im)
  (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+
  (* (- (* x.re x.re) (* x.im x.im)) x.im)
  (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -8 \cdot 10^{+153}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.7 \cdot 10^{+151}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -8e+153)
   (* 3.0 (* x.re (* x.re x.im)))
   (if (<= x.re 1.7e+151)
     (* x.im (- (* (* x.re x.re) 3.0) (* x.im x.im)))
     (* x.re (* (* x.re x.im) 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -8e+153) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_re <= 1.7e+151) {
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-8d+153)) then
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    else if (x_46re <= 1.7d+151) then
        tmp = x_46im * (((x_46re * x_46re) * 3.0d0) - (x_46im * x_46im))
    else
        tmp = x_46re * ((x_46re * x_46im) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -8e+153) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_re <= 1.7e+151) {
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -8e+153:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	elif x_46_re <= 1.7e+151:
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im))
	else:
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -8e+153)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_re <= 1.7e+151)
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) * 3.0) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -8e+153)
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_re <= 1.7e+151)
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	else
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -8e+153], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.7e+151], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -8e153

   1. Initial program 57.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 57.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 57.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 57.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 57.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 57.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 57.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 17.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 17.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 17.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 17.9%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 17.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 17.9%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 17.9%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 17.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 57.0%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 57.0%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 57.0%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 57.0%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 57.0%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 57.0%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 57.0%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 57.0%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 57.0%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 57.0%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.im around 0 63.9%

      \[\leadsto \color{blue}{3 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   10. Step-by-step derivation

       1. unpow2 63.9%

          \[\leadsto 3 \cdot \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im\right) \]

       2. associate-*l* 89.5%

          \[\leadsto 3 \cdot \color{blue}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]

   11. Simplified 89.5%

       \[\leadsto \color{blue}{3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]

   if -8e153 < x.re < 1.7e151

   1. Initial program 92.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 92.2%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 92.2%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 92.2%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 92.2%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 92.2%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 92.2%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 67.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 45.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 45.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 45.4%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 50.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 50.2%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 50.2%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 72.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 99.7%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 99.7%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 99.7%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 99.7%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 99.7%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 99.7%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 99.7%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 99.7%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 99.7%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 99.7%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   if 1.7e151 < x.re

   1. Initial program 49.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 49.4%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 49.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 49.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 49.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 49.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 49.4%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 16.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 16.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 16.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 16.9%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 16.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 16.9%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 16.9%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 16.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 49.4%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 49.4%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 49.4%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 49.4%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 49.4%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 49.4%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 49.4%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 49.4%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 49.4%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 49.4%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.im around 0 65.1%

      \[\leadsto \color{blue}{3 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   10. Step-by-step derivation

       1. unpow2 65.1%

          \[\leadsto 3 \cdot \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im\right) \]

       2. associate-*r* 65.2%

          \[\leadsto \color{blue}{\left(3 \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im} \]

       3. associate-*r* 65.1%

          \[\leadsto \color{blue}{\left(\left(3 \cdot x.re\right) \cdot x.re\right)} \cdot x.im \]

       4. associate-*l* 89.4%

          \[\leadsto \color{blue}{\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)} \]

       5. *-commutative 89.4%

          \[\leadsto \color{blue}{\left(x.re \cdot x.im\right) \cdot \left(3 \cdot x.re\right)} \]

       6. associate-*r* 89.2%

          \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)} \]

       7. *-commutative 89.2%

          \[\leadsto x.re \cdot \color{blue}{\left(\left(3 \cdot x.re\right) \cdot x.im\right)} \]

       8. associate-*l* 89.4%

          \[\leadsto x.re \cdot \color{blue}{\left(3 \cdot \left(x.re \cdot x.im\right)\right)} \]

   11. Simplified 89.4%

       \[\leadsto \color{blue}{x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -8 \cdot 10^{+153}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.7 \cdot 10^{+151}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \]

Alternative 2: 96.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (+
       (* x.im (- (* x.re x.re) (* x.im x.im)))
       (* x.re (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (fma (* x.im (* x.re 2.0)) x.re (* (* x.im (+ x.re x.im)) (- x.re x.im)))
   (* x.im (* x.im (- x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = fma((x_46_im * (x_46_re * 2.0)), x_46_re, ((x_46_im * (x_46_re + x_46_im)) * (x_46_re - x_46_im)));
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = fma(Float64(x_46_im * Float64(x_46_re * 2.0)), x_46_re, Float64(Float64(x_46_im * Float64(x_46_re + x_46_im)) * Float64(x_46_re - x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$im * N[(x$46$re * 2.0), $MachinePrecision]), $MachinePrecision] * x$46$re + N[(N[(x$46$im * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 92.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 92.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 92.3%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 92.3%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 92.3%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 92.3%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 92.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 92.3%

         \[\leadsto \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im} \]

      2. associate-*r* 92.4%

         \[\leadsto \color{blue}{\left(x.im \cdot \left(2 \cdot x.re\right)\right) \cdot x.re} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      3. fma-def 92.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(2 \cdot x.re\right), x.re, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im\right)} \]

      4. *-commutative 92.5%

         \[\leadsto \mathsf{fma}\left(x.im \cdot \color{blue}{\left(x.re \cdot 2\right)}, x.re, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im\right) \]

      5. *-commutative 92.5%

         \[\leadsto \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]

      6. difference-of-squares 92.5%

         \[\leadsto \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, x.im \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\right) \]

      7. associate-*r* 99.8%

         \[\leadsto \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \color{blue}{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)} \]

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 0.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 0.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 0.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 31.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 31.0%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 31.0%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 31.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 48.3%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 48.3%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 48.3%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 48.3%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 48.3%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 48.3%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 48.3%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 48.3%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 48.3%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 48.3%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.re around 0 72.4%

      \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 72.4%

          \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

       2. mul-1-neg 72.4%

          \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

       3. distribute-rgt-neg-in 72.4%

          \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   11. Simplified 72.4%

       \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 3: 77.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -8.2 \cdot 10^{+18} \lor \neg \left(x.re \leq 4.2 \cdot 10^{-26}\right):\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -8.2e+18) (not (<= x.re 4.2e-26)))
   (* 3.0 (* (* x.re x.re) x.im))
   (* x.im (* x.im (- x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -8.2e+18) || !(x_46_re <= 4.2e-26)) {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-8.2d+18)) .or. (.not. (x_46re <= 4.2d-26))) then
        tmp = 3.0d0 * ((x_46re * x_46re) * x_46im)
    else
        tmp = x_46im * (x_46im * -x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -8.2e+18) || !(x_46_re <= 4.2e-26)) {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -8.2e+18) or not (x_46_re <= 4.2e-26):
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im)
	else:
		tmp = x_46_im * (x_46_im * -x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -8.2e+18) || !(x_46_re <= 4.2e-26))
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -8.2e+18) || ~((x_46_re <= 4.2e-26)))
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	else
		tmp = x_46_im * (x_46_im * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -8.2e+18], N[Not[LessEqual[x$46$re, 4.2e-26]], $MachinePrecision]], N[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -8.2e18 or 4.20000000000000016e-26 < x.re

   1. Initial program 64.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around inf 66.3%

      \[\leadsto \color{blue}{{x.re}^{2}} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   3. Step-by-step derivation

      1. unpow2 66.3%

         \[\leadsto \color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   4. Simplified 66.3%

      \[\leadsto \color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. *-commutative 66.3%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]

      2. *-commutative 66.3%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]

      3. count-2 66.3%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)} \]

      4. associate-*l* 66.3%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + x.re \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.im\right)} \]

      5. *-commutative 66.3%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + x.re \cdot \left(\color{blue}{\left(x.re \cdot 2\right)} \cdot x.im\right) \]

      6. associate-*l* 66.3%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + x.re \cdot \color{blue}{\left(x.re \cdot \left(2 \cdot x.im\right)\right)} \]

      7. associate-*l* 66.3%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right)} \]

      8. distribute-lft-in 66.2%

         \[\leadsto \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(x.im + 2 \cdot x.im\right)} \]

      9. *-commutative 66.2%

         \[\leadsto \color{blue}{\left(x.im + 2 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right)} \]

      10. distribute-rgt1-in 66.2%

          \[\leadsto \color{blue}{\left(\left(2 + 1\right) \cdot x.im\right)} \cdot \left(x.re \cdot x.re\right) \]

      11. metadata-eval 66.2%

          \[\leadsto \left(\color{blue}{3} \cdot x.im\right) \cdot \left(x.re \cdot x.re\right) \]

      12. associate-*l* 66.3%

          \[\leadsto \color{blue}{3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)} \]

   6. Applied egg-rr 66.3%

      \[\leadsto \color{blue}{3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)} \]

   if -8.2e18 < x.re < 4.20000000000000016e-26

   1. Initial program 99.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 99.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 99.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 76.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 56.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 76.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 99.8%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 99.8%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 99.8%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 99.8%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 99.8%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 99.8%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.re around 0 91.1%

      \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 91.1%

          \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

       2. mul-1-neg 91.1%

          \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

       3. distribute-rgt-neg-in 91.1%

          \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   11. Simplified 91.1%

       \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -8.2 \cdot 10^{+18} \lor \neg \left(x.re \leq 4.2 \cdot 10^{-26}\right):\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 4: 82.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{+30} \lor \neg \left(x.re \leq 4.7 \cdot 10^{-26}\right):\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -2e+30) (not (<= x.re 4.7e-26)))
   (* 3.0 (* x.re (* x.re x.im)))
   (* x.im (* x.im (- x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2e+30) || !(x_46_re <= 4.7e-26)) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-2d+30)) .or. (.not. (x_46re <= 4.7d-26))) then
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    else
        tmp = x_46im * (x_46im * -x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2e+30) || !(x_46_re <= 4.7e-26)) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else {
		tmp = x_46_im * (x_46_im * -x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -2e+30) or not (x_46_re <= 4.7e-26):
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	else:
		tmp = x_46_im * (x_46_im * -x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -2e+30) || !(x_46_re <= 4.7e-26))
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -2e+30) || ~((x_46_re <= 4.7e-26)))
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	else
		tmp = x_46_im * (x_46_im * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -2e+30], N[Not[LessEqual[x$46$re, 4.7e-26]], $MachinePrecision]], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -2e30 or 4.69999999999999989e-26 < x.re

   1. Initial program 64.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 64.6%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 64.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 64.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 64.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 64.6%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 64.6%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 33.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 20.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 20.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 20.4%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 27.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 27.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 27.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 27.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 40.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 75.4%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 75.4%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 75.4%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 75.4%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 75.4%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 75.4%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 75.4%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 75.4%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 75.4%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 75.4%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.im around 0 66.3%

      \[\leadsto \color{blue}{3 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   10. Step-by-step derivation

       1. unpow2 66.3%

          \[\leadsto 3 \cdot \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im\right) \]

       2. associate-*l* 79.1%

          \[\leadsto 3 \cdot \color{blue}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]

   11. Simplified 79.1%

       \[\leadsto \color{blue}{3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]

   if -2e30 < x.re < 4.69999999999999989e-26

   1. Initial program 99.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 99.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 99.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 76.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 56.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 76.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 99.8%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 99.8%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 99.8%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 99.8%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 99.8%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 99.8%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.re around 0 91.1%

      \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 91.1%

          \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

       2. mul-1-neg 91.1%

          \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

       3. distribute-rgt-neg-in 91.1%

          \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   11. Simplified 91.1%

       \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{+30} \lor \neg \left(x.re \leq 4.7 \cdot 10^{-26}\right):\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 5: 82.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.85 \cdot 10^{+18}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.9 \cdot 10^{-27}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -2.85e+18)
   (* 3.0 (* x.re (* x.re x.im)))
   (if (<= x.re 1.9e-27)
     (* x.im (* x.im (- x.im)))
     (* x.re (* (* x.re x.im) 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -2.85e+18) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_re <= 1.9e-27) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-2.85d+18)) then
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    else if (x_46re <= 1.9d-27) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = x_46re * ((x_46re * x_46im) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -2.85e+18) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_re <= 1.9e-27) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -2.85e+18:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	elif x_46_re <= 1.9e-27:
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -2.85e+18)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_re <= 1.9e-27)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -2.85e+18)
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_re <= 1.9e-27)
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -2.85e+18], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.9e-27], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.85e18

   1. Initial program 65.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 65.4%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 65.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 65.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 65.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 65.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 65.4%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 25.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 18.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 18.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 18.6%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 24.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 24.2%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 24.2%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 24.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 30.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 76.6%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 76.6%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 76.6%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 76.6%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 76.6%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 76.6%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 76.6%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 76.6%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 76.6%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 76.6%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.im around 0 65.5%

      \[\leadsto \color{blue}{3 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   10. Step-by-step derivation

       1. unpow2 65.5%

          \[\leadsto 3 \cdot \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im\right) \]

       2. associate-*l* 79.3%

          \[\leadsto 3 \cdot \color{blue}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]

   11. Simplified 79.3%

       \[\leadsto \color{blue}{3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]

   if -2.85e18 < x.re < 1.9e-27

   1. Initial program 99.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 99.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 99.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 76.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 56.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 76.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 99.8%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 99.8%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 99.8%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 99.8%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 99.8%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 99.8%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.re around 0 91.1%

      \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 91.1%

          \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

       2. mul-1-neg 91.1%

          \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

       3. distribute-rgt-neg-in 91.1%

          \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   11. Simplified 91.1%

       \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   if 1.9e-27 < x.re

   1. Initial program 64.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 63.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 63.9%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 63.9%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 63.9%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 63.9%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 63.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 40.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 21.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 21.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 21.6%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 29.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 29.5%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 29.5%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 48.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 74.5%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 74.5%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 74.5%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 74.5%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 74.5%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 74.5%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 74.5%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 74.5%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 74.5%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 74.5%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.im around 0 66.8%

      \[\leadsto \color{blue}{3 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   10. Step-by-step derivation

       1. unpow2 66.8%

          \[\leadsto 3 \cdot \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im\right) \]

       2. associate-*r* 66.8%

          \[\leadsto \color{blue}{\left(3 \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im} \]

       3. associate-*r* 66.9%

          \[\leadsto \color{blue}{\left(\left(3 \cdot x.re\right) \cdot x.re\right)} \cdot x.im \]

       4. associate-*l* 79.0%

          \[\leadsto \color{blue}{\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)} \]

       5. *-commutative 79.0%

          \[\leadsto \color{blue}{\left(x.re \cdot x.im\right) \cdot \left(3 \cdot x.re\right)} \]

       6. associate-*r* 78.8%

          \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)} \]

       7. *-commutative 78.8%

          \[\leadsto x.re \cdot \color{blue}{\left(\left(3 \cdot x.re\right) \cdot x.im\right)} \]

       8. associate-*l* 78.9%

          \[\leadsto x.re \cdot \color{blue}{\left(3 \cdot \left(x.re \cdot x.im\right)\right)} \]

   11. Simplified 78.9%

       \[\leadsto \color{blue}{x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.85 \cdot 10^{+18}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.9 \cdot 10^{-27}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \]

Alternative 6: 82.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.2 \cdot 10^{+25}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 3.5 \cdot 10^{-29}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -2.2e+25)
   (* 3.0 (* x.re (* x.re x.im)))
   (if (<= x.re 3.5e-29)
     (* x.im (* x.im (- x.im)))
     (* (* x.re x.im) (* x.re 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -2.2e+25) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_re <= 3.5e-29) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-2.2d+25)) then
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    else if (x_46re <= 3.5d-29) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = (x_46re * x_46im) * (x_46re * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -2.2e+25) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_re <= 3.5e-29) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -2.2e+25:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	elif x_46_re <= 3.5e-29:
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -2.2e+25)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_re <= 3.5e-29)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -2.2e+25)
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_re <= 3.5e-29)
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -2.2e+25], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 3.5e-29], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.2000000000000001e25

   1. Initial program 65.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 65.4%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 65.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 65.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 65.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 65.4%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 65.4%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 25.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 18.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 18.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 18.6%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 24.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 24.2%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 24.2%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 24.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 30.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 76.6%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 76.6%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 76.6%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 76.6%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 76.6%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 76.6%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 76.6%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 76.6%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 76.6%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 76.6%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.im around 0 65.5%

      \[\leadsto \color{blue}{3 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   10. Step-by-step derivation

       1. unpow2 65.5%

          \[\leadsto 3 \cdot \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im\right) \]

       2. associate-*l* 79.3%

          \[\leadsto 3 \cdot \color{blue}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]

   11. Simplified 79.3%

       \[\leadsto \color{blue}{3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]

   if -2.2000000000000001e25 < x.re < 3.4999999999999997e-29

   1. Initial program 99.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 99.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

      2. *-commutative 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

      3. unpow2 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      4. associate-*r* 99.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

   4. Simplified 99.8%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 76.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

      2. expm1-udef 56.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

      3. +-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

      4. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

      5. distribute-lft-out 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

      6. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

      7. *-commutative 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 76.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

      2. expm1-log1p 99.8%

         \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

      3. unpow2 99.8%

         \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

      4. associate-+r- 99.8%

         \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

      5. associate-*r* 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      6. unpow2 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      7. *-commutative 99.8%

         \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

      8. distribute-lft1-in 99.8%

         \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

      9. metadata-eval 99.8%

         \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

      10. unpow2 99.8%

          \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

   9. Taylor expanded in x.re around 0 91.1%

      \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 91.1%

          \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

       2. mul-1-neg 91.1%

          \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

       3. distribute-rgt-neg-in 91.1%

          \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   11. Simplified 91.1%

       \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   if 3.4999999999999997e-29 < x.re

   1. Initial program 64.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around inf 66.9%

      \[\leadsto \color{blue}{{x.re}^{2}} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   3. Step-by-step derivation

      1. unpow2 66.9%

         \[\leadsto \color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   4. Simplified 66.9%

      \[\leadsto \color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. *-commutative 66.9%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]

      2. *-commutative 66.9%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + x.re \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]

      3. count-2 66.9%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + x.re \cdot \color{blue}{\left(2 \cdot \left(x.re \cdot x.im\right)\right)} \]

      4. associate-*l* 66.9%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + x.re \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.im\right)} \]

      5. *-commutative 66.9%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + x.re \cdot \left(\color{blue}{\left(x.re \cdot 2\right)} \cdot x.im\right) \]

      6. associate-*l* 66.9%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + x.re \cdot \color{blue}{\left(x.re \cdot \left(2 \cdot x.im\right)\right)} \]

      7. associate-*l* 66.8%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right)} \]

      8. distribute-lft-in 66.7%

         \[\leadsto \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(x.im + 2 \cdot x.im\right)} \]

      9. distribute-rgt1-in 66.7%

         \[\leadsto \left(x.re \cdot x.re\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot x.im\right)} \]

      10. metadata-eval 66.7%

          \[\leadsto \left(x.re \cdot x.re\right) \cdot \left(\color{blue}{3} \cdot x.im\right) \]

      11. associate-*r* 66.8%

          \[\leadsto \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot x.im} \]

      12. associate-*r* 66.9%

          \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re \cdot 3\right)\right)} \cdot x.im \]

      13. *-commutative 66.9%

          \[\leadsto \left(x.re \cdot \color{blue}{\left(3 \cdot x.re\right)}\right) \cdot x.im \]

      14. *-commutative 66.9%

          \[\leadsto \color{blue}{\left(\left(3 \cdot x.re\right) \cdot x.re\right)} \cdot x.im \]

      15. associate-*l* 79.0%

          \[\leadsto \color{blue}{\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)} \]

      16. *-commutative 79.0%

          \[\leadsto \color{blue}{\left(x.re \cdot 3\right)} \cdot \left(x.re \cdot x.im\right) \]

      17. *-commutative 79.0%

          \[\leadsto \left(x.re \cdot 3\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)} \]

   6. Applied egg-rr 79.0%

      \[\leadsto \color{blue}{\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.2 \cdot 10^{+25}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 3.5 \cdot 10^{-29}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \]

Alternative 7: 59.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ x.im \cdot \left(x.im \cdot \left(-x.im\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.im (* x.im (- x.im))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_im * -x_46_im);
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = x_46im * (x_46im * -x_46im)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_im * -x_46_im);
}

Python

def code(x_46_re, x_46_im):
	return x_46_im * (x_46_im * -x_46_im)

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_im * (x_46_im * -x_46_im);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

2. Taylor expanded in x.re around 0 81.9%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
3. Step-by-step derivation

   1. associate-*r* 81.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

   2. *-commutative 81.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

   3. unpow2 81.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

   4. associate-*r* 81.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

4. Simplified 81.9%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 54.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)\right)} \]

   2. expm1-udef 38.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)\right)} - 1} \]

   3. +-commutative 38.1%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} - 1 \]

   4. *-commutative 38.1%

      \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right)} - 1 \]

   5. distribute-lft-out 41.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right)} - 1 \]

   6. *-commutative 41.6%

      \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(\color{blue}{x.re \cdot \left(2 \cdot x.re\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

   7. *-commutative 41.6%

      \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot 2\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1 \]

6. Applied egg-rr 41.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 58.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)\right)\right)} \]

   2. expm1-log1p 87.4%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

   3. unpow2 87.4%

      \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re \cdot 2\right) + \left(\color{blue}{{x.re}^{2}} - x.im \cdot x.im\right)\right) \]

   4. associate-+r- 87.4%

      \[\leadsto x.im \cdot \color{blue}{\left(\left(x.re \cdot \left(x.re \cdot 2\right) + {x.re}^{2}\right) - x.im \cdot x.im\right)} \]

   5. associate-*r* 87.4%

      \[\leadsto x.im \cdot \left(\left(\color{blue}{\left(x.re \cdot x.re\right) \cdot 2} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

   6. unpow2 87.4%

      \[\leadsto x.im \cdot \left(\left(\color{blue}{{x.re}^{2}} \cdot 2 + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

   7. *-commutative 87.4%

      \[\leadsto x.im \cdot \left(\left(\color{blue}{2 \cdot {x.re}^{2}} + {x.re}^{2}\right) - x.im \cdot x.im\right) \]

   8. distribute-lft1-in 87.4%

      \[\leadsto x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot {x.re}^{2}} - x.im \cdot x.im\right) \]

   9. metadata-eval 87.4%

      \[\leadsto x.im \cdot \left(\color{blue}{3} \cdot {x.re}^{2} - x.im \cdot x.im\right) \]

   10. unpow2 87.4%

       \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]

8. Simplified 87.4%

   \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]

9. Taylor expanded in x.re around 0 55.7%

   \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
10. Step-by-step derivation

    1. unpow2 55.7%

       \[\leadsto x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

    2. mul-1-neg 55.7%

       \[\leadsto x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

    3. distribute-rgt-neg-in 55.7%

       \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

11. Simplified 55.7%

    \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

12. Final simplification 55.7%

    \[\leadsto x.im \cdot \left(x.im \cdot \left(-x.im\right)\right) \]

Alternative 8: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 -3.0)

C

double code(double x_46_re, double x_46_im) {
	return -3.0;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -3.0d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return -3.0;
}

Python

def code(x_46_re, x_46_im):
	return -3.0

Julia

function code(x_46_re, x_46_im)
	return -3.0
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = -3.0;
end

Wolfram

code[x$46$re_, x$46$im_] := -3.0

Derivation

1. Initial program 81.9%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

2. Taylor expanded in x.re around 0 81.9%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{2 \cdot \left({x.re}^{2} \cdot x.im\right)} \]
3. Step-by-step derivation

   1. associate-*r* 81.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(2 \cdot {x.re}^{2}\right) \cdot x.im} \]

   2. *-commutative 81.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(2 \cdot {x.re}^{2}\right)} \]

   3. unpow2 81.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \left(2 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

   4. associate-*r* 81.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.im \cdot \color{blue}{\left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]

4. Simplified 81.9%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right)} \]
5. Step-by-step derivation

   1. +-commutative 81.9%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(2 \cdot x.re\right) \cdot x.re\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im} \]

   2. associate-*r* 81.9%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(2 \cdot x.re\right)\right) \cdot x.re} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

   3. fma-def 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(2 \cdot x.re\right), x.re, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im\right)} \]

   4. *-commutative 83.9%

      \[\leadsto \mathsf{fma}\left(x.im \cdot \color{blue}{\left(x.re \cdot 2\right)}, x.re, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im\right) \]

   5. *-commutative 83.9%

      \[\leadsto \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]

   6. difference-of-squares 87.1%

      \[\leadsto \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, x.im \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\right) \]

   7. associate-*r* 93.6%

      \[\leadsto \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \color{blue}{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

6. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 2\right), x.re, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)} \]

7. Taylor expanded in x.im around 0 51.9%

   \[\leadsto \color{blue}{\left(2 \cdot {x.re}^{2} + {x.re}^{2}\right) \cdot x.im} \]

9. Simplified 2.7%

   \[\leadsto \color{blue}{-3} \]

10. Final simplification 2.7%

    \[\leadsto -3 \]

Developer target: 91.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023171 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))